+15 Let x and y be nonnegative real numbers. If x^2+3y^2=18, then find the maximum value of xy.
0
As we need to find the max value of a product we can use AM-GM Inequality. the inequality states that for any real numbers \(x_1, x_2, \ldots, x_n \geq 0\),
\(\LARGE\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}\)
with equality if and only if \(x_1 = x_2 = \cdots = x_n\)
Thus using this we write:
\(\Large\frac{x^2+3y^2}{2}\ge\sqrt[2]{x^2\cdot3y^2}\)
\(\)Plugging in values and solving-->
\(\sqrt3xy\le9\\ \boxed{xy\le3\sqrt3}\)
0
As we need to find the max value of a product we can use AM-GM Inequality. the inequality states that for any real numbers \(x_1, x_2, \ldots, x_n \geq 0\),
\(\LARGE\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}\)
with equality if and only if \(x_1 = x_2 = \cdots = x_n\)
Thus using this we write:
\(\Large\frac{x^2+3y^2}{2}\ge\sqrt[2]{x^2\cdot3y^2}\)
\(\)Plugging in values and solving-->
\(\sqrt3xy\le9\\ \boxed{xy\le3\sqrt3}\)
